Formulating Cointegrated Series as Coupled Stochastic Differential Equations

Summary

The document proposes a two-variable stochastic differential equation model for related time series. Each series has a drift term tied to a linear combination of both variables, alongside its own Brownian noise. The model includes a correlation parameter for the two noise processes and assumes the drift coefficients have opposite signs, reflecting a possible restoring relationship between the series.

The author asks whether analytical solutions for both series can be found, including the case of uncorrelated noise, and seeks references. No derivation, solution, or empirical evidence is provided, so the document is a modeling question rather than a worked method. It may be useful as a starting point for studying linear systems of stochastic differential equations and cointegration, but readers must establish the solution conditions and connection to statistical cointegration independently.

Key ideas

• The proposed model couples two stochastic series through their drift terms.
• Each series is driven by Brownian noise, and the model allows the noise terms to be correlated.
• The author assumes the two drift coefficients have opposite signs.
• The document asks about analytical solutions but does not provide one.

Tags

# Simultaneous Stochastic Differential Equations


# Simultaneous Stochastic Differential Equations












I was thinking about cointegrated time series and came up with the following simultaneous equations model:

$dY_t = \alpha (Y_t - \gamma X_t)dt + \sigma dB_t$

$dX_t = \beta (Y_t - \delta X_t)dt + \tau dW_t$

$dW_t dB_t = \rho dt$

With greek letters constants. $\alpha$ and $\beta$ with opposite signs.

Is it possible to find analytical solutions for $Y_t$ and $X_t$ (maybe allowing for $\rho = 0$) ?

I looked in Oksendal and Shreve (and obviously google) for technics to solve it, but couldn't find a clue. Any references would be appreciated.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.